Dynamical systems
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Dynamical Systems — scope drawn from open graduate dynamical systems notes and typical US graduate dynamical systems syllabi.
Flows and invariant sets
- Existence — Existence
- Uniqueness for ODEs — Uniqueness for ODEs
- Flows, orbits, and invariant sets — Flows, orbits, and invariant sets
- Limit sets, ω-limit sets, and attractors — Limit sets, ω-limit sets, and attractors
- Lyapunov stability — Lyapunov stability
- Lyapunov functions — Lyapunov functions
- Stable and unstable manifolds (local theory) — Stable and unstable manifolds (local theory)
Bifurcation theory
- Parameter-dependent equilibria — Parameter-dependent equilibria; structural stability (introduction)
- Saddle-node and transcritical bifurcations — Saddle-node and transcritical bifurcations
- Pitchfork bifurcations — Pitchfork bifurcations
- Hopf bifurcation — Hopf bifurcation
- Limit cycle creation — Limit cycle creation
- Normal forms — Normal forms
- Center manifold reduction (introduction) — Center manifold reduction (introduction)
- Codimension-two bifurcations (overview) — Codimension-two bifurcations (overview)
Discrete and symbolic dynamics
- Diffeomorphisms and return maps — Diffeomorphisms and return maps
- Smale horseshoe and symbolic dynamics — Smale horseshoe and symbolic dynamics
- Topological entropy (introduction) — Topological entropy (introduction)
- Period doubling cascade and universality — Period doubling cascade and universality
- Hamiltonian dynamics — Hamiltonian dynamics
- KAM theory (overview) — KAM theory (overview)
Applied dynamical systems
- Pattern formation in PDEs (reaction–diffusion, introduction) — Pattern formation in PDEs (reaction–diffusion, introduction)
- Celestial mechanics — Celestial mechanics
- N-body problems (overview) — N-body problems (overview)
- Neuronal dynamics and bursting models — Neuronal dynamics and bursting models
- Epidemiological — Epidemiological
- Ecological models with bifurcations — Ecological models with bifurcations
- Control of nonlinear systems near equilibria — Control of nonlinear systems near equilibria
Computation and analysis tools
- Numerical continuation — Numerical continuation
- Bifurcation software (AUTO, MatCont overview) — Bifurcation software (AUTO, MatCont overview)
- Lyapunov exponent computation — Lyapunov exponent computation
- Delay differential equations (introduction) — Delay differential equations (introduction)
- Random dynamical systems (introduction) — Random dynamical systems (introduction)
- Data-driven reconstruction of attractors (overview) — Data-driven reconstruction of attractors (overview)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Flows and invariant sets
Existence
Coming soon - Bifurcation theory
Parameter-dependent equilibria; structural stability (introduction)
Coming soon - Discrete and symbolic dynamics
Diffeomorphisms and return maps
Coming soon - Applied dynamical systems
Pattern formation in PDEs (reaction–diffusion, introduction)
Coming soon - Computation and analysis tools
Numerical continuation
Coming soon
Notes
Graduate Dynamical Systems — scope drawn from open graduate dynamical systems notes and typical US graduate dynamical systems syllabi. Topic outline: `content/topics/graduate/dynamical_systems.json`.